Conjugacy class

Classe de conjugaison (Fr). Konjugiertenklasse (Ge). Classe coniugata (It). 共役類 (Ja). Класс сопряжённости (Ru). Clase de conjugación (Sp).

If g₁ and g₂ are two elements of a group G, they are called conjugate if there exists an element g₃ in G such that:

g₃g₁g₃⁻¹ = g₂.

Conjugacy is an equivalence relation and therefore partitions G into equivalence classes: every element of the group belongs to precisely one conjugacy class.

The equivalence class that contains the element g₁ in G is

Cl(g₁) = { g₃g₁g₃⁻¹| g₃ ∈ G }

and is called the conjugacy class of g₁. The class number of G is the number of conjugacy classes.

The classes Cl(g₁) and Cl(g₂) are equal if and only if g₁ and g₂ are conjugate, and disjoint otherwise.

For Abelian groups the concept is trivial, since each element forms a class on its own.